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A237590 a(n) is the total number of regions (or parts) after n-th stage in the diagram of the symmetries of sigma described in A236104.

%I A237590 #52 Dec 31 2020 11:11:15
%S A237590 1,2,4,5,7,8,10,11,14,16,18,19,21,23,26,27,29,30,32,33,37,39,41,42,45,
%T A237590 47,51,52,54,55,57,58,62,64,67,68,70,72,76,77,79,80,82,84,87,89,91,92,
%U A237590 95,98,102,104,106,107,111,112,116,118,120,121,123,125,130,131,135,136,138,140,144,147,149,150,152,154
%N A237590 a(n) is the total number of regions (or parts) after n-th stage in the diagram of the symmetries of sigma described in A236104.
%C A237590 The total area (or total number of cells) of the diagram after n stages is equal to A024916(n), the sum of all divisors of all positive integers <= n.
%C A237590 Note that the region between the virtual circumscribed square and the diagram is a symmetric polygon whose area is equal to A004125(n), see example.
%C A237590 For more information see A237593 and A237270.
%C A237590 a(n) is also the total number of terraces of the stepped pyramid with n levels described in A245092. - _Omar E. Pol_, Apr 20 2016
%H A237590 Robert Price, <a href="/A237590/b237590.txt">Table of n, a(n) for n = 1..5000</a>
%H A237590 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr01.jpg">An infinite stepped pyramid</a>
%H A237590 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr02.jpg">Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)</a>
%H A237590 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the stepped pyramid (first 16 levels)</a>
%F A237590 a(n) = A317109(n) - A294723(n) + 1 (Euler's formula). - _Omar E. Pol_, Jul 21 2018
%e A237590 Illustration of initial terms:
%e A237590 .                                                         _ _ _ _
%e A237590 .                                           _ _ _        |_ _ _  |_
%e A237590 .                               _ _ _      |_ _ _|       |_ _ _|   |_
%e A237590 .                     _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
%e A237590 .             _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
%e A237590 .       _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
%e A237590 .      |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
%e A237590 .
%e A237590 .
%e A237590 .       1      2        4          5            7              8
%e A237590 .
%e A237590 For n = 6 the diagram contains 8 regions (or parts), so a(6) = 8.
%e A237590 The sum of all divisors of all positive integers <= 6 is [1] + [1+2] + [1+3] + [1+2+4] + [1+5] + [1+2+3+6] = 33. On the other hand after 6 stages the sum of all parts of the diagram is [1] + [3] + [2+2] + [7] + [3+3] + [12] = 33, equaling the sum of all divisors of all positive integers <= 6.
%e A237590 Note that the region between the virtual circumscribed square and the diagram is a symmetric polygon whose area is equal to A004125(6) = 3.
%e A237590 From _Omar E. Pol_, Dec 25 2020: (Start)
%e A237590 Illustration of the diagram after 29 stages (contain 215 vertices, 268 edges and 54 regions or parts):
%e A237590 ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e A237590 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e A237590 |_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
%e A237590 |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
%e A237590 |_ _ _ _ _ _ _ _ _ _ _ _ _  | |
%e A237590 |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
%e A237590 |_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
%e A237590 |_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
%e A237590 |_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
%e A237590 |_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
%e A237590 |_ _ _ _ _ _ _ _ _ _  | |       |_ _|   |_
%e A237590 |_ _ _ _ _ _ _ _ _ _| | |_ _    |_  |_ _  |_ _
%e A237590 |_ _ _ _ _ _ _ _ _  | |_ _ _|     |_  | |_ _  |
%e A237590 |_ _ _ _ _ _ _ _ _| | |_ _  |_      |_|_ _  | |
%e A237590 |_ _ _ _ _ _ _ _  | |_ _  |_ _|_        | | | |_ _ _ _ _ _
%e A237590 |_ _ _ _ _ _ _ _| |     |     | |_ _    | |_|_ _ _ _ _  | |
%e A237590 |_ _ _ _ _ _ _  | |_ _  |_    |_  | |   |_ _ _ _ _  | | | |
%e A237590 |_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
%e A237590 |_ _ _ _ _ _  | |_  |_  |_    | |_|_ _ _ _  | | | | | | | |
%e A237590 |_ _ _ _ _ _| |_ _|   |_  |   |_ _ _ _  | | | | | | | | | |
%e A237590 |_ _ _ _ _  |     |_ _  | |_ _ _ _  | | | | | | | | | | | |
%e A237590 |_ _ _ _ _| |_      | |_|_ _ _  | | | | | | | | | | | | | |
%e A237590 |_ _ _ _  |_ _|_    |_ _ _  | | | | | | | | | | | | | | | |
%e A237590 |_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
%e A237590 |_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
%e A237590 |_ _ _|   |_ _  | | | | | | | | | | | | | | | | | | | | | |
%e A237590 |_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
%e A237590 |_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
%e A237590 |_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
%e A237590 |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
%e A237590 .
%e A237590 (End)
%t A237590 (* total number of parts in the first n symmetric representations *)
%t A237590 (* Function a237270[] is defined in A237270 *)
%t A237590 (* variable "previous" represents the sum from 1 through m-1 *)
%t A237590 a237590[previous_,{m_,n_}]:=Rest[FoldList[Plus[#1,Length[a237270[#2]]]&,previous,Range[m,n]]]
%t A237590 a237590[n_]:=a237590[0,{1,n}]
%t A237590 a237590[78] (* data *)
%t A237590 (* _Hartmut F. W. Hoft_, Jul 07 2014 *)
%Y A237590 Partial sums of A237271.
%Y A237590 Compare with A060831 (analog for the diagram that contains subparts).
%Y A237590 Cf. A000203, A004125, A024916, A196020, A236104, A235791, A237048, A237270, A237591, A237593, A239659, A239660, A239663, A239665, A239931-A239934, A245092, A244050, A244970, A262626, A317109.
%K A237590 nonn
%O A237590 1,2
%A A237590 _Omar E. Pol_, Mar 31 2014
%E A237590 Definition clarified by _Omar E. Pol_, Jul 21 2018